PermPal Database

Av(132, 4231, 4321)

Generating Function

\frac{x^{4} - x^{3} + 4 x^{2} - 3 x + 1}{{(x - 1)}^{4}}

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Counting Sequence

1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452, 585, 742, ...

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Implicit Equation for the Generating Function

{(x - 1)}^{4} F (x) - x^{4} + x^{3} - 4 x^{2} + 3 x - 1 = 0

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (3) = 5

a (4) = 12

a (n) = \frac{n (n^{2} - 3 n + 5)}{3}, n \geq 5

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Explicit Closed Form

{\begin{array}{cc} 1 & n = 0 \\ \frac{n (n^{2} - 3 n + 5)}{3} & otherwise \end{array}

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This specification was found using the strategy pack "Point Placements" and has 30 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

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+ Expand

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\begin{aligned} F_{0} (x) & = F_{1} (x) + F_{2} (x) \\ F_{1} (x) & = 1 \\ F_{2} (x) & = F_{3} (x) \\ F_{3} (x) & = F_{4} (x) F_{5} (x) \\ F_{4} (x) & = x \\ F_{5} (x) & = F_{6} (x) + F_{9} (x) \\ F_{6} (x) & = F_{1} (x) + F_{7} (x) \\ F_{7} (x) & = F_{8} (x) \\ F_{8} (x) & = F_{4} (x) F_{6} (x) \\ F_{9} (x) & = F_{10} (x) + F_{17} (x) \\ F_{10} (x) & = F_{11} (x) \\ F_{11} (x) & = F_{12} (x) F_{4} (x) \\ F_{12} (x) & = F_{13} (x) + F_{6} (x) \\ F_{13} (x) & = F_{14} (x) + F_{7} (x) \\ F_{14} (x) & = F_{15} (x) \\ F_{15} (x) & = F_{16} (x) F_{4} (x) \\ F_{16} (x) & = F_{14} (x) + F_{7} (x) \\ F_{17} (x) & = F_{18} (x) + F_{19} (x) + F_{29} (x) \\ F_{18} (x) & = 0 \\ F_{19} (x) & = F_{20} (x) F_{4} (x) \\ F_{20} (x) & = F_{16} (x) + F_{21} (x) \\ F_{21} (x) & = F_{22} (x) + F_{25} (x) \\ F_{22} (x) & = F_{15} (x) + F_{18} (x) + F_{23} (x) \\ F_{23} (x) & = F_{24} (x) F_{4} (x) \\ F_{24} (x) & = F_{22} (x) + F_{7} (x) \\ F_{25} (x) & = F_{26} (x) \\ F_{26} (x) & = F_{27} (x) F_{4} (x) \\ F_{27} (x) & = F_{14} (x) + F_{28} (x) \\ F_{28} (x) & = F_{26} (x) \\ F_{29} (x) & = F_{4} (x) F_{9} (x) \end{aligned}

Av(132, 4231, 4321)

This specification was found using the strategy pack "Point Placements" and has 30 rules.

Proof Tree

System of Equations